In [1]}, the following expansion is given for \(s=\alpha +it\) , \(|t| \le 1/(u^{1/3}\log y)\) :
\(\frac{\zeta (s,y) x^{s}}{s} = \frac{\zeta (\alpha ,y)x^{\alpha }}{\alpha }e^{-\sigma _2 t^2/2} \left(1-\frac{it}{\alpha }-i\frac{t^3}{6}\sigma _3 + O\left( t^6 \sigma _3^2 + t^2 \alpha ^{-2}+t^4 \sigma _4\right)\right)\) 
